Posts Tagged ‘Nash equilibrium’

Together with Formula 1, tennis is the other sport I love – and my favourite player by far is Rafa!

We often apply game theory to various sports and consider how players, teams and individuals can think strategically. One of the big debates in tennis is ‘who is the best ever’ and I argue that Nadal is the greatest, based on a huge range of metrics.

I saw this article in the Economist, providing analysis and comparison between some of the best tennis players. It shows how we can use economic thinking, probability, game theory and analysis to come to something of an answer about who is the greatest, considering the various players’ runs to the title in the Grand Slams. Of course the reason I’m posting this is because according to the Economist, Rafa is the best! And the reasoning is very sound. Enjoy. I certainly did.

People are beginning to get used to low oil prices and acting as if they are going to remain low. Oil is trading at only a little over $30 per barrel and Saudi Arabia is unwilling to backtrack on its policy of maintaining its level of production and not seeking to prevent oil prices from falling. Currently, there is still a position of over supply and hence in the short term the price could continue falling – perhaps to $20 per barrel.

But what of the future? What will happen in the medium term (6 to 12 months) and the longer term? Investment in new oil wells, both conventional and shale oil, have declined substantially. The position of over supply could rapidly come to an end. The Telegraph article below quotes the International Energy Agency’s executive director, Fatih Birol, as saying:

“Investment in oil exploration and production across the world has been cut to the bone, falling 24% last year and an estimated 17% this year. This is… far below the minimum levels needed to keep up with future demand. …

It is easy for consumers to be lulled into complacency by ample stocks and low prices today, but they should heed the writing on the wall: the historic investment cuts raise the odds of unpleasant oil security surprises in the not too distant future.”

In today’s oil market there is hardly any spare production capacity other than in Saudi Arabia and Iran and significant investment is required just to maintain existing production before we move on to provide the new capacity needed to meet rising oil demand. The risk of a sharp oil price rise towards the later part of our forecast arising from insufficient investment is as potentially de-stabilising as the sharp oil price fall has proved to be.

The higher-cost conventional producers, such as Venezuela, Nigeria, Angola, Russia and off-shore producers, could take a long time to rebuild capacity as investment in conventional wells is costly, especially off-shore.

As far as shale oil producers is concerned – the prime target of Saudi Arabia’s policy of not cutting back supply – production could well bounce back after a relatively short time as wells are re-opened and investment in new wells is resumed.

But, price rises in the medium term could then be followed by lower prices again a year or two thereafter as oil from new investment comes on stream: or they could continue rising if investment is insufficient. It depends on the overall balance of demand and supply. The table shows the IEA’s forecast of production and consumption and the effect on oil stocks. From 2018, it is predicting that consumption will exceed production and that, therefore, stocks will fall – and at an accelerating rate.

But just what happens to the balance of production and consumption will also depend on expectations. If shale oil investors believe that an oil price bounce is temporary, they are likely to hold off investing. But this will, in turn, help to sustain a price bounce, which in turn, could help to encourage investment. So expectations of investors will depend on what other investors expect to happen – a very difficult outcome to predict. It’s a form of Keynesian beauty contest (see the blog post A stock market beauty contest of the machines) where what is important is what other people think will happen, which in turn depends on what they think other people will do, and so on.

After two weeks of negotiations between the 195 countries attending the COP21 climate change conference in Paris, a deal has been reached on tackling climate change. Although the deal still has to be ratified by countries, this is a major step forward in limiting global warming. Before it can formally come into force, it must have been ratified by at least 55 countries, accounting for at least 55% of global greenhouse gas emissions.

The deal goes much further than previous agreements and includes the following:

•

A limit on the increase in global temperatures to ‘well below’ 2°C above pre-industrial levels and efforts pursued to limit it to 1.5°C.

•

A recognition that the pledges already made ahead of the conference by 186 countries and incorporated into the agreement are insufficient and will only limit global temperature rise to 2.7°C at best.

•

Countries to update their emissions reductions commitments every five years – the first being in 2020. Such revised commitments should then be legally binding.

•

A global ‘stocktake’ in 2023, and every five years thereafter, to monitor countries’ progress in meeting their commitments and to encourage them to make deeper cuts in emissions to reach the 1.5°C goal. This requires a process of measurement and verification of countries’ emissions.

•

To reach a peak in greenhouse gas emissions as soon as possible and then to begin reducing them and to achieve a balance between sources and sinks of greenhouse gases (i.e. zero net emissions) in the second half of this century.

•

Developed countries to provide the poorest developing countries with $100bn per year by 2020 to help them reduce emissions. This was agreed in Copenhagen, but will now be continued from 2020 to 2025, and by 2025 a new goal above $100bn per year will be agreed.

•

The development of market mechanisms that would award tradable credits for green projects and emissions reductions.

•

A recognition that the ‘loss and damage’ associated with climate-related disasters can be serious for many vulnerable developing countries (such as low-lying island states) and that this may require compensation. However, there is no legal liability on developed countries to provide such compensation.

Perhaps the major achievement at the conference was a universal recognition that the problem of global warming is serious and that action needs to be taken. Mutual self interest was the driving force in reaching the agreement, and although it is less binding on countries than many would have liked, it does mark a significant step forward in tackling climate change.

But why did the conference not go further? Why, if there was general agreement that global warming should be tackled and that global temperature rise should ideally be capped at 1.5°C, was there not a binding agreement on each country to apply this cap?

There are two reasons.

First, it is very difficult to predict the exact relationship, including its timing, between emissions and global temperature rise. Even if you could make limits to emissions binding, you could not make global temperature rise binding.

Second, even if there is general agreement about how much emissions should be reduced, there is no general agreement on the distribution of these reductions. Many countries want to do less themselves and others to do more. More specifically, poor countries want rich countries to do all the cutting while many continue to build more coal-fired power stations to provide the electricity to power economic development. The rich countries want the developing countries, especially the larger ones, such as China, India and Brazil to reduce their emissions, or at least the growth in their emissions.

Then there is the difference between what countries vaguely pledge at a global conference and what they actually do domestically. Many developed countries are keen to take advantage of currently cheap fossil fuels to power economic growth. They are also still investing in alternative sources of fossil fuels, such as through fracking.

As we said in the previous blog, game theory can shed some useful insights into the nature and outcome of climate negotiations. ‘The global optimum may be for a strong agreement, binding on all countries. The Nash equilibrium, however, may be a situation where countries push for their own interests at the expense of others, with the final agreement being much more minimalistic.’

‘Minimalistic’ may be too strong a description of the outcomes of the Paris conference. But they could have been stronger. Nevertheless, judged by the outcomes of previous climate conferences, the deal could still be described as ‘historic’.

The Paris Climate Change Conference (COP21) is under way. At the opening on November 30, 150 Heads of State gathered in Paris, most of whom addressed the conference. With representatives from 195 countries and observers from a range of organisations, the conference is set to last until 11 December. Optimism is relatively high that a legally binding and universal agreement will be reached, with the aim of keeping global warming below 2°C – what is generally regarded as a ‘safe’ limit.

But although it is hoped that a successor to the Kyoto Protocol of 1997 will be put in place, there are many problems in getting so many countries to agree. They may all wish to reduce global warming, but there is disagreement on how it should be achieved and how the burden should be shared between countries.

There are several difficult economic issues in the negotiations. The first is the size and impact of the external costs of emissions. When a country burns fossil fuels, the benefits are almost entirely confined to residents of that county. However, the environmental costs are largely external to that country and only a relatively small fraction is borne by that country and hardly at all by the polluters themselves, unless there is a carbon tax or other form or penalty in place. The problem is that the atmosphere is a common resource and without collective action – national or international – it will be overused.

The second problem is one of distribution. Politicians may agree in principle that a solution is necessary which is equitable between nations, but there is considerable disagreement on what is meant by ‘equitable’ in this context. As the third Guardian article below puts it:

The most important hurdle could be over whether industrialised countries like the US, UK and Japan, which have contributed the most to the historical build-up of emissions, should be obliged to cut more than developing countries. India, on behalf of many poor countries, will argue that there must be “differentiation” between rich and poor; but the US wants targets that are applicable to all. A collision is inevitable.

A third problem is that of uncertainty. Although there is general agreement among scientists that human action is contributing to global warming, there is less agreement on the precise magnitude of the causal relationships. There is also uncertainty over the likely effects of specific emissions reductions. This uncertainty can then be used by governments which are unwilling to commit too much to emissions reductions.

A fourth difficulty arises from the intertemporal distribution of costs and benefits of emissions reductions. The costs are born immediately action is taken. Carbon taxes or charges, or subsidies to renewables, or caps on emissions, all involve higher energy prices and/or higher taxes. The flows of benefits (or lower costs), however, of reduced emissions are not likely to be fully experienced for a very long time. But governments, whether democratic or dictatorships, tend to have a relatively short time horizon, governed by the electoral cycle or the likelihood of staying in power. True, governments may not be solely concerned with power and many politicians may have genuine desires to tackle climate change, but their political survival is still likely to be a major determinant of their actions.

Of course, if there is strong public opinion in favour of action to reduce emissions, governments are likely to respond to this. Indeed, all the expressions of public support for action ahead of the conference from all around the world, do give some hope for a strong agreement at the Paris conference. Nevertheless, there is still widespread scepticism in many countries over the relationship between human action and climate change, and many argue that the costs of policies to tackle climate change exceed the benefits.

Game theory can shed some insights into the difficulties ahead for the negotiators. The global optimum may be for a strong agreement, binding on all countries. The Nash equilibrium, however, may be a situation where countries push for their own interests at the expense of others, with the final agreement being much more minimalistic.

There do, however, seem to be more reasons to be cheerful at this summit that at previous ones. But negotiations are likely to be hard and protracted over the coming days.

John Nash was one of the pioneers of game theory and in 1994 was awarded the Nobel prize in economics for his work in this field. He was also the subject of the 2001 film, A beautiful mind, where the young John Nash was played by Russell Crowe, for which he won an Oscar.

Tragically, John Nash and his wife were killed in a car crash on May 23: he was 86 years old. Since his death there have been many tributes paid to him and his work.

As a student of economics you will almost certainly have studied the concept of a Nash equilibrium: a situation where everyone makes their best choice, given the choices of the other ‘players’ in a ‘game’. But a Nash equilibrium is not in the collective best interests of the participants in a non-cooperative game.

This is a very different concept of equilibrium from the simple equilibrium in a perfectly competitive market. In fact, in the complex world of strategic decision making, where firms are constantly looking at their rivals’ behaviour and possible reactions to their own behaviour, there are all sorts of Nash equilibria which are clearly sub-optimal. Competition may be highly destructive.

The following obituaries look at Nash’s contribution to the development of economics. As the Bloomberg article states:

The game theory concepts that Nash’s math[s] brought to the field were a true paradigm shift in economics. Macroeconomists, who continue to use the old Walrasian notion of equilibrium typically engage in hand-waving about how the macroeconomy is too big for strategic interactions to matter. But most of the economics profession has gradually shifted toward Nash equilibrium. The 2014 Nobel winner, Jean Tirole, is emblematic of the new economics. And many of the biggest successes in applied economics, like the auctions that power Google’s advertising, rely on Nash’s technique.

Each day many investors anxiously watch the stock market to see if their shares have gone up or down. They may also speculate: buying if they think share prices are likely to go up; selling if they think their shares will fall. But what drives these expectations?

To some extent, people will look at real factors, such as company sales and profits or macroeconomic indicators, such as the rate of economic growth or changes in public-sector borrowing. But to a large extent people are trying to predict what other people will do: how other people will react to changes in various indicators.

John Maynard Keynes observed this phenomenon in Chapter 12 of his General Theory of Employment, Interest and Money of 1936. He likened this process of anticipating what other people will do to a newspaper beauty contest, popular at the time. In fact, behaviour of this kind has become known as a Keynesian beauty contest (see also). Keynes wrote that:

professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one’s judgement, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees.

When investors focus on people’s likely reactions, it can make markets very unstable. A relatively minor piece of news can cause people to buy or sell in anticipation that others will do the same and that others will realise this and do the same themselves. Markets can overshoot, until, when prices have got out of line with fundamentals, buying can turn into selling, or vice versa. Prices can then move rapidly in the other direction, again driven by what people think other people will do. Sometimes, markets can react to very trivial news indeed. As the New York Times article below states:

On days without much news, the market is simply reacting to itself. And because anxiety is running high, investors make quick, sometimes impulsive, responses to relatively minor events.

The rise of the machineIn recent years there is a new factor to account for growing stock market volatility. The Keynesian beauty contest is increasingly being played by computers. They are programmed to buy and sell when certain conditions are met. The hundreds of human traders of the past who packed trading floors of stock markets, have been largely replaced by just a few programmers, trained to adjust the algorithms of the computers their finance companies use as trading conditions change.

And these computers react in milliseconds to what other computers are doing, which in turn react to what others are doing. Markets can, as a result, suddenly soar or plummet, until the algorithms kick the market into reverse as computers sell over-priced stock or buy under-priced stock, which triggers other computers to do the same.

Robot trading is here to stay. The articles and podcast consider the implications of the ‘games’ they are playing – for savers, companies and the economy.

John Von Neumann was a mathematician and one of his many accolades was applying mathematics and his observations of traditional games to create a new discipline – Game Theory. This involves a mathematical approach to decision making whereby different strategies can be assessed. It a tool that not only can be used in Economics, but also can be applied to a broad range of areas and fields of study.

Just as I arrived at work, I was listening to Radio 4 and heard the introduction to the programme In our Time. This one in particular caught my attention because of the name mentioned – Von Neumann, and after arriving in my office I then listened to the discussions surrounding game theory.

The main link is to the discussion from BBC Radio 4, led by Melvyn Bragg, with guests: Ian Stewart, a Professor of Mathematics from the University of Warwick; Andrew Colman, a Professor of Psychology at the University of Leicester and Richard Bradley, a Professor of Philosophy from the LSE. I’ll keep it brief and simply say enjoy!

At the end of two weeks of often acrimonious wrangling between representatives from 193 countries, an agreement – of sorts – was reached at the climate change summit in Copenhagen. What was this agreement? It was an ‘accord’ brokered by the USA, China, India, Brazil and South Africa.

This Copenhagen Accord contains three elements. The first is a recognition of the need to prevent global temperatures rising by more than 2 degrees Celsius above pre-industrial levels. The second is a commitment by developed countries to give $30bn of aid between 2010 and 2012 to developing countries for investment in green technology and to mitigate the effects of climate change. In addition, a goal was set of providing $100bn a year by 2020. The third is for rich countries to give pledges on emissions reductions and for developing countries to give pledges on reducing emissions increases. Developed countries’ pledges will be scrutinised by the UN Framework Convention on Climate Change, while developing countries will merely be required to submit reports on their progress in meeting their pledges.

But this is only an accord. It has no legal status and was merely ‘recognised’ by the countries at the conference. What is more, the target of limiting temperature rises to 2C does not contain a date by which temperature rises should peak. Also, as countries are not required to submit targets for emissions until February 2010, it is not clear how these targets will be kept low enough to meet the temperature target and there is no identification of penalites that would apply to countries not meeting their pledges.

Not surprisingly, reactions around the world have been mixed. The following podcasts and articles look at these reactions and at the economic mechanisms that will be required to meet the 2C limit